{"paper":{"title":"Generalised Flatness Constants: A Framework Applied in Dimension $2$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO","math.SG"],"primary_cat":"math.MG","authors_text":"Giulia Codenotti, Johannes Hofscheier, Thomas Hall","submitted_at":"2021-09-21T15:34:21Z","abstract_excerpt":"Let $A \\in \\{ \\mathbb{Z}, \\mathbb{R} \\}$ and $X \\subset \\mathbb{R}^d$ be a bounded set. Affine transformations given by an automorphism of $\\mathbb{Z}^d$ and a translation in $A^d$ are called (affine) $A$-unimodular transformations. The image of $X$ under such a transformation is called an $A$-unimodular copy of $X$. It was shown in [Averkov, Hofscheier, Nill, 2019] that every convex body whose width is \"big enough\" contains an $A$-unimodular copy of $X$. The threshold when this happens is called the generalised flatness constant $\\mathrm{Flt}_d^A(X)$. It resembles the classical flatness const"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2110.02770","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2110.02770/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}